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Mirrors > Home > ILE Home > Th. List > rmo5 | GIF version |
Description: Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.) |
Ref | Expression |
---|---|
rmo5 | ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mo 2030 | . 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))) | |
2 | df-rmo 2463 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
3 | df-rex 2461 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
4 | df-reu 2462 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
5 | 3, 4 | imbi12i 239 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))) |
6 | 1, 2, 5 | 3bitr4i 212 | 1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∃wex 1492 ∃!weu 2026 ∃*wmo 2027 ∈ wcel 2148 ∃wrex 2456 ∃!wreu 2457 ∃*wrmo 2458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 df-mo 2030 df-rex 2461 df-reu 2462 df-rmo 2463 |
This theorem is referenced by: nrexrmo 2693 cbvrmo 2702 bdrmo 14544 |
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